Wilson loops: from pseudo-holomorphic surfaces to 2d YM Riccardo Ricci Imperial College London Cambridge, 12 Nov Based on 0704.2237, 0707.2699, 0711.3226 and 0905.0665 with N. Drukker, S. Giombi, V. Pestun and D. Trancanelli Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM
Outline Wilson loops in N = 4 SYM Review of 1 / 2 BPS loops: straight line and circle Zarembo’s construction of supersymmetric Wilson loops 1 / 16 BPS Wilson loops on S 3 General construction Examples AdS string theory dual: Wilson loops as pseudoholomorphic surfaces Relation to 2d Yang-Mills (YM 2 ) Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM
Introduction and motivation In this talk I will discuss a new family of supersymmetric Wilson loop operators in N = 4 SYM theory Field content: A µ , Φ I , Ψ A I = 1 , . . . , 6 , A = 1 , . . . , 4 Symmetry group: PSU (2 , 2 | 4) ⊃ SO (5 , 1) × SU (4) R + 32 SUSY’s (16 Q + 16 S ) This gauge theory is believed to be dual to type IIB string theory on AdS 5 × S 5 . This is a strong/weak duality which is in general very hard to test directly. Interesting tests of the conjecture can be obtained by considering certain subsectors of the theory which preserve some fractions of the supersymmetries of the vacuum. Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM
Introduction and motivation Supersymmetric local operators are well studied both in the gauge theory and in AdS , but less is known about non-local operators. One motivation is then the classification of supersymmetric non-local operators such as Wilson loops. � W [ C ] ∼ Tr R P exp i A + · · · C Wilson loops are described in AdS by strings (and D-branes): probe the duality beyond the strict supergravity limit. Certain supersymmetric Wilson loops may provide exact results interpolating between weak and strong coupling. Remarkable connection with scattering amplitudes Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM
Part I Field theory analysis Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM
Wilson loops in N = 4 SYM In N = 4 SYM it is natural to define � W = 1 x µ ( s ) + | ˙ x | Θ I ( s )Φ I N Tr R P exp ds ( iA µ ˙ � ) � �� Coupling to N = 4 scalars - x µ ( s ): loop on R 4 I = 1 , . . . , 6 Θ 2 = 1 - Θ I ( s ): loop on S 5 , All such loops are locally supersymmetric � x | Θ I ρ I � x µ γ µ + | ˙ δ W ∝ i ˙ ǫ 0 ǫ 0 : 16 comp. constant spinor � �� � Squares to zero (because Θ 2 = 1) So the susy equation δ W = 0 has solutions, but in general only point by point along the loop. Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM
We will be interested in operators which preserve fractions of the full 32 supersymmetries of N = 4 SYM globally . � ǫ 0 : 16 Q ’s ǫ ( x ) = ǫ 0 + x µ γ µ ǫ 1 ǫ 1 : 16 S ’s The susy variation can then be written as � x | Θ I ρ I � x µ γ µ + | ˙ δ W ∝ i ˙ ǫ ( x ) the Wilson loop will be supersymmetric if we can find solutions for constant ǫ 0 , ǫ 1 independent of the point along the loop . Problem: how to choose ( x µ ( s ) , Θ I ( s )) so that W is supersymmetric? Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM
1 / 2 BPS straight line x 2 x µ = ( s , 0 , 0 , 0) Θ I = (1 , . . . , 0) Couples to a single scalar Φ 1 x 1 It is easy to see that susy variation gives γ 1 − i ρ 1 � γ 1 − i ρ 1 � � � ǫ 0 = 0 , ǫ 1 = 0 → Preserves 8 Q ’s and 8 S ’s, so it is 1/2 BPS It has trivial expectation value to all orders in λ , N � W � = 1 Checked both in perturbation theory and at strong coupling in AdS . Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM
1 / 2 BPS circle x 2 x µ = (cos s , sin s , 0 , 0) Θ I = (1 , . . . , 0) x 1 Again it couples to only one of the scalars Now in the susy variation ǫ 0 and ǫ 1 are not decoupled ρ 1 ǫ 0 = i γ 12 ǫ 1 → It is also 1/2 BPS, but preserves 16 combinations of Q and S . The VEV is non trivial � W � = f ( λ, N ) and remarkably it is exactly computable! Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM
1 / 2 BPS circle and matrix model The VEV for the 1 / 2 BPS circle is fully captured by the Hermitean gaussian matrix model: Erikson, Semenoff, Zarembo ‘00, Drukker-Gross ‘00 � � � � W � = 1 D φ 1 − 2 N N Tr e φ exp λ Tr φ 2 λ = g 2 , YM N Z Crucial observation: the combined gauge-scalar propagator between two points on the loop is a constant = g 2 �� �� �� iA + φ i A a ( x ) + Φ a ( x ) i A b ( y ) + Φ b ( y ) 8 π 2 δ ab YM = → The sum of all non-interacting graphs (“ladder diagrams”) is equal to the matrix model! Interacting graphs must vanish. Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM
The matrix model can be exactly solved and yields the prediction � � − g 2 g 2 � W � = 1 N L 1 YM YM e 8 N − 1 4 � � √ 1 + λ � 8 + . . . λ ≪ 1 2 √ At large N : � W � = I 1 λ ∼ √ λ λ ≫ 1 λ e Large λ limit agrees with string calculation in AdS √ e − S F 1 = e λ Certain 1 / N corrections were also successfully reproduced by D-branes (dual to Wilson loops in large representations). Drukker, Fiol ’05 The MM/half-BPS loop relation can be proven using localization techniques. Pestun ’07 Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM
Zarembo’s supersymmetric Wilson loops Can we find interesting generalizations of the 1 / 2 BPS Wilson loops? Given a curve x µ ( s ), how to choose the scalar couplings Θ I ( s ) so that some supersymmetry is preserved? Two possible directions: i Generalize the straight line loop Zarembo ’02 ii Generalize the circular loop Given an arbitrary curve x µ ( s ) ∈ R 4 and four scalars Φ µ define � W = 1 Θ µ = ˙ x µ A µ + ˙ x µ Φ µ ) , x µ / | ˙ N Tr P exp ds ( i ˙ x | The loop dependence drops out in the supersymmetry variation ( γ µ − i ρ µ ) ǫ 0 = 0 4 indep. projectors → generically only 1 Q is preserved. Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM
An interesting fact is that the VEV for these loops is always trivial to all orders in λ , N � W � = 1 as can be checked both in perturbation theory and in AdS . Dymarsky, Gubser, Guralnik and Maldacena ’06 Generalization of straight line loop as � W straight line � = 1 Unfortunately not much to compute with them! Clearly this family of operators does not contain the 1/2 BPS circle, which is a very interesting observable. Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM
The loops on S 3 In Zarembo’s construction we select four of the scalars Φ µ and couple them to the 1-forms dx µ . The construction of susy loops on S 3 is similar, but the basic ingredient to define the scalar couplings are now the left invariant one forms of SU (2) = S 3 . Defining U = i τ i x i + I x 4 , these are σ L = U † dU = σ i , L i τ i 2 , i = 1 , 2 , 3 In cartesian coordinates, they may be explicitly written as σ i , L = 2 σ i x µ x µ = 1 µν x µ dx ν , where σ i µν are constant matrices: σ i σ i 4 j = δ i jk = ǫ ijk , j Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM
The loops on S 3 Given an arbitrary curve x µ ( s ) on a unit radius S 3 , we can couple the three one-forms σ L i to three of the scalars. Our definition for susy Wilson loops on S 3 is then � � � W = 1 i A + 1 2 σ L i Φ i N Tr P exp , i = 1 , 2 , 3 It is a locally supersymmetric operator: Θ i ( s ) ds = 1 x | → Θ 2 = 1 2 σ L i / | ˙ Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM
Supersymmetry Now we show that our definition of Wilson loop on S 3 leads to a supersymmetric operator: � x µ x ν ρ i � x µ γ µ − σ i ( ǫ 0 + x η γ η ǫ 1 ) = 0 δ W ∝ i ˙ µν ˙ An important point that allows to solve this equation is that the left invariant one forms are related to the action of the Lorentz group on right (anti-chiral) spinors ǫ − : γ µν ǫ − = i σ i µν τ i ǫ − Similarly we have γ µν ǫ + = i ˜ µν τ i ǫ + , σ R = UdU † σ i Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM
Supersymmetry A generic curve on S 3 is supersymmetric if ( τ i + ρ i ) ǫ − ǫ − 1 = − ǫ − ǫ + 1 = ǫ + 0 = 0 , 0 , 0 = 0 � �� � set chiral spinors to zero for i = 1 , 2 , 3. This leaves 2 out of the 8 components of ǫ − 0 . Since ǫ − 1 is related to ǫ − 0 our Wilson loops preserve 2 combinations of ¯ Q and ¯ S The loops are 1/16 BPS Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM
Examples: Great circle We have constructed a new infinite family of supersymmetric Wilson loops. For an arbitrary curve on S 3 , they are 1 / 16 BPS. But for special shapes one can have enhanced supersymmetry. We recover the 1 / 2 BPS circle: it is a great circle of the S 3 ! S 3 Loop: x µ = (cos s , sin s , 0 , 0) 2 = 0 , σ L σ L 1 = σ L Scalar couplings: 2 = 1 3 Familiar 1/2 BPS circle coupled to a single scalar As for the 1 / 2 BPS circle, all loops in this new family have non trivial VEVs. Are they exactly computable? Riccardo Ricci Wilson loops: from pseudo-holomorphic surfaces to 2d YM
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